176 research outputs found
Response Time is More Important than Walking Speed for the Ability of Older Adults to Avoid a Fall after a Trip
We previously reported that the probability of an older adult recovering from a forward trip and using a “lowering” strategy increases with decreased walking velocity and faster response time. To determine the within-subject interaction of these variables we asked three questions: (1) Is the body orientation at the time that the recovery foot is lowered to the ground (“tilt angle”) critical for successful recovery? (2) Can a simple inverted pendulum model, using subject-specific walking velocity and response time as input variables, predict this body orientation, and thus success of recovery? (3) Is slower walking velocity or faster response time more effective in preventing a fall after a trip? Tilt angle was a perfect predictor of a successful recovery step, indicating that the recovery foot placement must occur before the tilt angle exceeds a critical value of between 23° and 26° from vertical. The inverted pendulum model predicted the tilt angle from walking velocity and response time with an error of 0.4±2.2° and a correlation coefficient of 0.93. The model predicted that faster response time was more important than slower walking velocity for successful recovery. In a typical individual who is at risk for falling, we predicted that a reduction of response time to a normal value allows a 77% increase in safe walking velocity. The mathematical model produced patient-specific recommendations for fall prevention, and suggested the importance of directing therapeutic interventions toward improving the response time of older adults
Germinal Center Selection and Affinity Maturation Require Dynamic Regulation of mTORC1 Kinase
During antibody affinity maturation, germinal center (GC) B cells cycle between affinity-driven selection in the light zone (LZ) and proliferation and somatic hypermutation in the dark zone (DZ). Although selection of GC B cells is triggered by antigen-dependent signals delivered in the LZ, DZ proliferation occurs in the absence of such signals. We show that positive selection triggered by T cell help activates the mechanistic target of rapamycin complex 1 (mTORC1), which promotes the anabolic program that supports DZ proliferation. Blocking mTORC1 prior to growth prevented clonal expansion, whereas blockade after cells reached peak size had little to no effect. Conversely, constitutively active mTORC1 led to DZ enrichment but loss of competitiveness and impaired affinity maturation. Thus, mTORC1 activation is required for fueling B cells prior to DZ proliferation rather than for allowing cell-cycle progression itself and must be regulated dynamically during cyclic re-entry to ensure efficient affinity-based selection
Cauchy's infinitesimals, his sum theorem, and foundational paradigms
Cauchy's sum theorem is a prototype of what is today a basic result on the
convergence of a series of functions in undergraduate analysis. We seek to
interpret Cauchy's proof, and discuss the related epistemological questions
involved in comparing distinct interpretive paradigms. Cauchy's proof is often
interpreted in the modern framework of a Weierstrassian paradigm. We analyze
Cauchy's proof closely and show that it finds closer proxies in a different
modern framework.
Keywords: Cauchy's infinitesimal; sum theorem; quantifier alternation;
uniform convergence; foundational paradigms.Comment: 42 pages; to appear in Foundations of Scienc
A Burgessian critique of nominalistic tendencies in contemporary mathematics and its historiography
We analyze the developments in mathematical rigor from the viewpoint of a
Burgessian critique of nominalistic reconstructions. We apply such a critique
to the reconstruction of infinitesimal analysis accomplished through the
efforts of Cantor, Dedekind, and Weierstrass; to the reconstruction of Cauchy's
foundational work associated with the work of Boyer and Grabiner; and to
Bishop's constructivist reconstruction of classical analysis. We examine the
effects of a nominalist disposition on historiography, teaching, and research.Comment: 57 pages; 3 figures. Corrected misprint
Symmetry of matrix-valued stochastic processes and noncolliding diffusion particle systems
As an extension of the theory of Dyson's Brownian motion models for the
standard Gaussian random-matrix ensembles, we report a systematic study of
hermitian matrix-valued processes and their eigenvalue processes associated
with the chiral and nonstandard random-matrix ensembles. In addition to the
noncolliding Brownian motions, we introduce a one-parameter family of
temporally homogeneous noncolliding systems of the Bessel processes and a
two-parameter family of temporally inhomogeneous noncolliding systems of Yor's
generalized meanders and show that all of the ten classes of eigenvalue
statistics in the Altland-Zirnbauer classification are realized as particle
distributions in the special cases of these diffusion particle systems. As a
corollary of each equivalence in distribution of a temporally inhomogeneous
eigenvalue process and a noncolliding diffusion process, a stochastic-calculus
proof of a version of the Harish-Chandra (Itzykson-Zuber) formula of integral
over unitary group is established.Comment: LaTeX, 27 pages, 4 figures, v3: Minor corrections made for
publication in J. Math. Phy
Leibniz's Infinitesimals: Their Fictionality, Their Modern Implementations, And Their Foes From Berkeley To Russell And Beyond
Many historians of the calculus deny significant continuity between
infinitesimal calculus of the 17th century and 20th century developments such
as Robinson's theory. Robinson's hyperreals, while providing a consistent
theory of infinitesimals, require the resources of modern logic; thus many
commentators are comfortable denying a historical continuity. A notable
exception is Robinson himself, whose identification with the Leibnizian
tradition inspired Lakatos, Laugwitz, and others to consider the history of the
infinitesimal in a more favorable light. Inspite of his Leibnizian sympathies,
Robinson regards Berkeley's criticisms of the infinitesimal calculus as aptly
demonstrating the inconsistency of reasoning with historical infinitesimal
magnitudes. We argue that Robinson, among others, overestimates the force of
Berkeley's criticisms, by underestimating the mathematical and philosophical
resources available to Leibniz. Leibniz's infinitesimals are fictions, not
logical fictions, as Ishiguro proposed, but rather pure fictions, like
imaginaries, which are not eliminable by some syncategorematic paraphrase. We
argue that Leibniz's defense of infinitesimals is more firmly grounded than
Berkeley's criticism thereof. We show, moreover, that Leibniz's system for
differential calculus was free of logical fallacies. Our argument strengthens
the conception of modern infinitesimals as a development of Leibniz's strategy
of relating inassignable to assignable quantities by means of his
transcendental law of homogeneity.Comment: 69 pages, 3 figure
On Poincare and logarithmic Sobolev inequalities for a class of singular Gibbs measures
This note, mostly expository, is devoted to Poincar{\'e} and log-Sobolev
inequalities for a class of Boltzmann-Gibbs measures with singular interaction.
Such measures allow to model one-dimensional particles with confinement and
singular pair interaction. The functional inequalities come from convexity. We
prove and characterize optimality in the case of quadratic confinement via a
factorization of the measure. This optimality phenomenon holds for all beta
Hermite ensembles including the Gaussian unitary ensemble, a famous exactly
solvable model of random matrix theory. We further explore exact solvability by
reviewing the relation to Dyson-Ornstein-Uhlenbeck diffusion dynamics admitting
the Hermite-Lassalle orthogonal polynomials as a complete set of
eigenfunctions. We also discuss the consequence of the log-Sobolev inequality
in terms of concentration of measure for Lipschitz functions such as maxima and
linear statistics.Comment: Minor improvements. To appear in Geometric Aspects of Functional
Analysis -- Israel Seminar (GAFA) 2017-2019", Lecture Notes in Mathematics
225
A Hexagon Model for 3D Lorentzian Quantum Cosmology
We formulate a dynamically triangulated model of three-dimensional Lorentzian
quantum gravity whose spatial sections are flat two-tori. It is shown that the
combinatorics involved in evaluating the one-step propagator (the transfer
matrix) is that of a set of vicious walkers on a two-dimensional lattice with
periodic boundary conditions and that the entropy of the model scales
exponentially with the volume. We also give explicit expressions for the
Teichm\"uller parameters of the spatial slices in terms of the discrete
parameters of the 3d triangulations, and reexpress the discretized action in
terms of them. The relative simplicity and explicitness of this model make it
ideally suited for an analytic study of the conformal-factor cancellation
observed previously in Lorentzian dynamical triangulations and of its relation
to alternative, reduced phase space quantizations of 3d gravity.Comment: 34 pages, 20 figures, some clarifying remarks added, final version to
appear in Phys Rev
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